Free-Dirac-particle evolution as a quantum random walk

Abstract

It is known that any positive-energy state of a free Dirac particle that is initially highly localized evolves in time by spreading at speeds close to the speed of light. As recently indicated by Strauch, this general phenomenon, and the resulting "two-horned" distributions of position probability along any axis through the point of initial localization, can be interpreted in terms of a quantum random walk, in which the roles of "coin" and "walker" are naturally associated with the spin and translational degrees of freedom in a discretized version of Dirac's equation. We investigate the relationship between these two evolutions analytically and show how the evolved probability density on the x axis for the Dirac particle at any time t can be obtained from the asymptotic form of the probability distribution for the position of a "quantum walker." The case of a highly localized initial state is discussed as an example.

Emergence of randomness and arrow of time in quantum walks

“…Here Tr c expresses the partial trace for the coin state. Physically speaking, the DTQW can express the free Dirac equation [25][26][27] as follows. We assume the specific quantum coin flip,…”

Relativistic effects in quantum walks: Klein's paradox and zitterbewegung

“…Despite their contribution to the quantum information processing, quantum walks are themselves very interesting physical systems worth being studied due to effects from various fields of physics like quantum chaos [5] and solid state physics [6]. Recently, it has been shown that discrete time quantum walk (DTQW) resemble the one-dimensional free particle Dirac equation [7,8]. Actually, one has to keep in mind that the idea of quantum walks goes back to Feynmann and Hibbs [9] who considered a discrete version of the one-dimensional Dirac equation propagator.…”

How to Realize One-dimensional Discrete-time Quantum Walk by Dirac Particle

“…This analysis is similar to Ref. 30 and can be extended to the two-dimensional space [32]. It is emphasized that the DTQW can be implemented under well-controlled nonrelativistic quantum mechanics while the DTQW can simulate the relativistic quantum mechanics.…”

“…Various DTQWs can simulate the Dirac equation [30][31][32][33][34], the spatially discretized Schrödinger equation [35,36], the Klein-Gordon equation [33,37], or other various differential equations [38,39]. Furthermore, classical dynamics can be simulated [40].…”